Question Video: Identifying the Correct Diagonal in the Parallelogram Law Mathematics

Which vector is equivalent to ๐ฎ + ๐ฏ?


Video Transcript

Which vector is equivalent to ๐ฎ plus ๐ฏ?

Looking at our figure, we see these two vectors making up the sides of a four-sided shape. Since the length of this side equals the magnitude of vector ๐ฏ and so does the length of this side, while these two sides of our shape have lengths equal to the magnitude of vector ๐ฎ, we know the shape overall is a parallelogram. Using this shape then, there are actually two ways to find our answer.

Starting at point ๐ด, we could follow along vector ๐ฎ and then after that vector ๐ฏ, and the tip of vector ๐ฏ points to where the sum of these two vectors lies. Here weโ€™re using whatโ€™s called the tip-to-tail method. Note that the tail of vector ๐ฏ lies at of the tip of vector ๐ฎ. Arranged this way, the sum of these two vectors ๐ฎ plus ๐ฏ is indicated by a vector that goes from the tail of the first vector, vector ๐ฎ, to the tip of the second vector, vector ๐ฏ.

We mentioned that thereโ€™s a second way to solve for ๐ฎ plus ๐ฏ. And our shape now shows us how. ๐ฎ plus ๐ฏ is equivalent to ๐ฏ plus ๐ฎ. So we could get the same result by starting at the tail of vector ๐ฏ and then where its tip meets the tail of vector ๐ฎ, following that vector to its end. This approach gives us the same result of vector ๐ฎ plus vector ๐ฏ. In terms of the corners of our parallelogram, we see that this resultant vector joins point ๐ด to point ๐ท. We can express a vector from point ๐ด to point ๐ท like this. This, then, is our final answer. Vector ๐ฎ plus vector ๐ฏ equals vector ๐€๐ƒ.

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